how to start and solve each question

1. Let there be two goods with quantities x = (x1.xz) E RE is 3:1 32 0, x2 2 0. Let prices 33 = (ppm) E R$+ i'e p1 > 0.33; > 0. Let the consumer's preferences be represented by the CES 1 utility function U(x1.x2) = (xf + x') I\" where-p 0. a. Find the long run total cost function C(w, r, y). (10) b. Let capital be xed at K = R in the short run. Find the short run total cost function C(w,r.y;K = E). (6} c. Let p be the price of v. If this is a competitive firm, find how much would it produce in the short run. [6] 3. Let the firm's output be 3: E R+. Let the output price be p E RH. Let the quantities ofthe m inputs be x = (x1, . xm) E El\". Let input prices be w = (wl, , wm) E R311. The rm takes input prices as given ie they are parameters. Prove that the profit function rr(p, w) is non- decreasing in p. (6) 4. Concavitv and quasiconcavitv a. Let f(x) = :l:JD forx E RH. Notice that it here is a single variable, x E RH, not a vector. Using calculus, determine for which values of the parameter p the function x9 is a concave function over the convex set R\". (4) b. Let f and g be concave functions over a convex set .S' E R". Let a, b E 8+ is a 2 0, b :2 0. Dene the function of + bg overS to be (of + bg )(x) = ax) + bg(x). Prove that the function af + by is a concave function over 51(6) General case: If f1. f2. f" are concave functions over the convex set .S' E R\" and a = (:11, 12.51\") E R3 then a1 1' 1 + 1sz + akf" is a concave function over the set 5'. Don't need to prove. Here f 2, f 3, etcjust means a second function, third function, etc, not the square of a function, cube, etc. Similar results apply to convex functions